You may also like

All strange numbers are prime. Every one digit prime number is
strange and a number of two or more digits is strange if and only
if so are the two numbers obtained from it by omitting either its
first or its last digit. Find all strange numbers.

Can you find a relationship between the number of dots on the
circle and the number of steps that will ensure that all points are
hit?

Sieve of Eratosthenes

Age 11 to 14 Challenge Level:

Why do this problem?

This problem offers students opportunities to explore
multiples in more depth than usual, in particular looking at the
links between multiples of different numbers. It also
encourages students to see the connection between primes and
multiples.

Possible approach

"What are the first few multiples of 2?"
"2, 4, 6, 8, 10, ..."

"And multiples of 7?"
"7, 14, 21, 28, 35, ..."

"Great. We'll be investigating properties of multiples
today."
[Hand out sheets of smaller
grids, one sheet per pair of students.]

"I'd like you to shade in all the multiples of 2 except 2, but
before you do that, turn to your neighbour and try to predict what patterns you'll
produce."
[Give them a minute to make predictions and do the
shading. Emphasise that there is no need for beautiful
shading.]

"I'd like you to shade in all the multiples of 3 except
3. Again, before you do that, turn to your neighbour and try
to predict what patterns
you'll produce."
[Give them a couple of minutes to do this.]

"Were your predictions correct? Why did you make those
predictions?
Can you explain why you get
different patterns for multiples of 2 and multiples of
3?"

"Now let's think about what happens when we combine these
multiples."
[Hand out master grid,
with multiples of 2 already crossed out.]
"We'll use this as our master grid to keep a running record of our
findings. It's already got the multiples of 2 crossed
out. Before you cross out the multiples of 3, can you and
your partner predict what
will happen? Will you cross out any numbers that are already
crossed out? If so, which ones?"
[Give them a couple of minutes to work on this, and then ask them
to report back.]

"What am I going to ask you to do next?"

"OK, so now explore what happens for multiples of 4, 5, 6 and
7. Before you shade in the multiples on the small grids, try
to predict what patterns
might emerge. After you've shaded in the multiples, try to explain the patterns you've
found.
Before you update the master grid, try to predict what will happen.
Will you cross out any numbers that are already crossed out?
If so, which ones?
After you've updated the master grid, try to explain why some numbers have
been crossed out again and others haven't."
[Give them a few minutes for this.]

"Now look at the master grid. What is special about the
numbers that you haven't crossed out?

"What would change on the master
grid if you were to cross out multiples of larger
numbers?"

"Imagine you want to find all the
prime numbers up to 400. You could do this by crossing out
multiples in a 2-400 number grid. Which multiples will
you choose to cross out? How can you be sure that you are
left with the primes?"[You might want to have some2-400 gridsavailable in case students would like to
try it.]

Key questions

Which numbers get crossed out more
than once, and why?Which numbers don't get crossed out
at all, and why?Which possible factors do we need
to consider in order to decide if a number is
prime?

Possible extension

"We're used to working with grids
with ten columns, but you might find an interesting result if you
use thissix-column gridinstead. Can you
predict what you will
see?"

Possible support

By working in pairs we are encouraging students to share ideas
and support each other.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice.